Wavelet transform is an important tool in signal processing, used in particular to perform time-frequency analyses. In general terms, a wavelet transform is described by the convolution:
                                          W            s                    ⁡                      (            τ            )                          =                              1                          s                                ⁢                                    ∫                              -                ∞                            ∞                        ⁢                                          x                ⁡                                  (                  t                  )                                            ⁢                                                ψ                  *                                ⁡                                  (                                                            t                      -                      τ                                        s                                    )                                            ⁢                                                          ⁢                              ⅆ                t                                                                        (        1        )                            where:                    t is time;            x(t) is the signal to be transformed;            function ψεL2() is the “mother wavelet”, also called the “generator function”; it generally presents a compact support [−α, +α];            s>0 is called the “scale factor”;            τ is the time delay;            “*” indicates the conjugate complex; in fact, ψ can generally be a function with complex values, but in the following we shall consider, without losing generality, the case where it is a real function;            Ws(τ) is a wavelet coefficient dependent on s and τ.                        
We talk about “Discrete Wavelet Transform” (DWT), when s=am and τ=nb, where m,n are positive integers. If s and τ can adopt any value—any positive value in the case of s—we talk about “Continuous Wavelet Transform” (CWT), even if signal x is discrete, which is generally the case, to allow it to be digitally processed.
DWTs are generally more suitable for processing neuronal signals, for example, because they allow said signals to be analyzed at the desired frequential resolution.
For more details on wavelet transform, one may refer to: Beatrice Pesquet-popescu and Jean-Christophe Pesquet “Ondelettes et applications” (“Wavelets and applications”), Techniques de l'Ingénieur, Chapter TE 5 215.
Fast algorithms have been developed for performing discrete wavelet transforms. In contrast, continuous wavelet transforms remain relatively onerous in terms of time and resources, which limits their use in real time applications and low energy consumption applications.
In a conventional manner, a CWT is implemented, in an approximate fashion, by replacing the integral of equation (1) by a discrete sum. The case of a signal x(t) sampled at discrete intervals t=−1, 0, 1, 2 . . . is considered (for the sake of simplicity, and without losing generality, the time scale is taken such that its discretization step equals 1), and xi=x(ti) is put down; the mother wavelet ψ is defined on a compact support [−α, +α]. The function:
            ψ      s        ⁡          (      t      )        =            1              s              ⁢          ψ      ⁡              (                  t          s                )            is defined, whose support is [−αs, +αs] where αs=α/s. Equation (1) then becomes:
            W      s        ⁡          (      τ      )        =                    ∫                  -                      α            s                                    α          s                    ⁢                        x          ⁡                      (                          t              +              τ                        )                          ⁢                              ψ            s                    ⁡                      (            t            )                          ⁢                                  ⁢                  ⅆ          t                      =                            ∑                      i            =                          -                              α                s                                                                          α            s                    -          1                    ⁢                        ∫          i                      i            +            1                          ⁢                              x            ⁡                          (                              t                +                τ                            )                                ⁢                                    ψ              s                        ⁡                          (              t              )                                ⁢                                          ⁢                      ⅆ            t                              
To calculate this sum of integrals, we make the piecewise constant signal approximation: x(t)=xi ∀tε[i, i+1[(other approximations are possible, for example, linear interpolation between t=i and t=i+1); furthermore, one limits oneself to integer values of τ(τ= . . . , −1, 0, 1, . . . ). One can therefore write:
                                                        W              s                        ⁡                          (              τ              )                                =                                                    ∑                                  i                  =                                      -                    α                                                                              α                -                1                                      ⁢                                          A                i                            ⁢                              x                                  i                  +                  τ                                                                    ⁢                                  ⁢                              with            ⁢                                                  ⁢                          A              i                                =                                    ∫              i                              i                +                1                                      ⁢                                                            ψ                  s                                ⁡                                  (                  t                  )                                            ⁢                                                          ⁢                                                ⅆ                  t                                .                                                                        (        2        )            
By considering the coefficients Ai as known, which can be calculated once and for all, one sees that determining a single wavelet coefficient requires (2α−1) multiplications and (2α−2) sums. Now, α can adopt relatively large values, for example, to make a time-frequency analysis of a signal sampled at 1 kHz in frequency range 10 Hz-300 Hz with Meyer wavelets, α is typically between 100 and 3000.
The article by M. Omachi and S. Omachi “Fast calculation of continuous wavelet transform using polynomial”, Proceedings of the 2007 International Conference on Wavelet Analysis and Pattern Recognition, Beijing, China, 2-4 Nov. 2007 describes a fast CWT method for a signal, in which the number of operations required does not depend on α. In this method, the mother wavelet is defined by a polynomial of order d on a compact support: typically, order d and the coefficients of the polynomial are chosen in such a way as to approximate, with the desired accuracy, a non-polynomial “reference” wavelet, such as a Haar wavelet. The wavelet coefficients are calculated in a recursive manner; hence Ws(τ) is calculated from Ws−1(τ), the scale factor being considered an integer. The computational cost of calculating coefficient Ws(τ) is O(ds).
This method has a certain number of disadvantages.
Firstly, the polynomial approximation of a “reference wavelet”, such as the Haar wavelet, may necessitate recourse to a high order d, which increases the computational cost of the method accordingly.
Secondly, it does not allow coefficients Ws to be determined for different values of s—and therefore for different frequencies—independently of each other. On the contrary, the recursive character of the algorithm means that the determination of Ws necessarily involves determining W1 . . . Ws−1. The method of M. Omachi and S. Omachi is therefore not optimized for the applications in which a signal has to be analyzed only in certain spectral bands.